I have learned that $\mathbf{C}$ is a parabolic Riemann surface. But I do not know how to show that :
(1) The punctured plane $\mathbf{C}^{*}$ is also a parabolic Riemann surface, but it is not when omits two points of $\mathbf{C}$.
(2)Moreover, if Riemann surface $\mathbf{M}$ is compact, then $$\mathbf{M} \backslash \{p_{1},...,p_{n}\}$$ is the parabolic Riemann surface.
(3)If Riemann surface $\mathbf{M}$, then $$\mathbf{M} \backslash B_{r}(z_{0}) $$ is the hyperbolic Riemann surface.